Boundary layers surface shear stress

When only approximate values of the overall features of the flow, such as the surface heat transfer rate and surface shearing stress, are required, it is possible to apply the boundary layer assumptions in a different way to obtain, relatively easily, approximate solutions for these quantities. The derivation of the equations required in this approximate solution procedure will be discussed in the present section (21,[7),[13]. 

On the other hand, at a free surface, where the fluid layer is bounded above (or below) by an inviscid fluid the boundary condition of shear-stress continuity is approximated as... 

The effects of the viscosity variation across the boundary layer on the surface shear stress and heat flux are shown in Fig. 6.7. The shear stress is normalized by the value obtained from the Bla-... 

For a given value of x and the same flow properties, the boundary layer on a cone (k = 1) is thinner by a factor of 1/V3, and the surface shear stress and heat transfer are larger by a factor of V3. The local skin friction and heat transfer coefficients are related similarly ... 

The parameter k is called the von Karman constant, and the value that fits most of the data is 0.41. The corresponding value of B is 5.0. Intermediate between these layers is the buffer layer, where both shear mechanisms are important. The essential feature of this data correlation is that the wall shear completely controls the turbulent boundary layer velocity distribution in the vicinity of the wall. So dominant is the effect of the wall shear that even when pressure gradients along the surface are present, the velocity distributions near the surface are essentially coincident with the data obtained on plates with uniform surface pressure [82]. Within this region for a flat plate, the local shear stress remains within about 10 percent of the surface shear stress. It is noted that this shear variation is often ignored in turbulent boundary layer theory. 

A better way to obtain u, would be a direct measurement of the surface shear stress, but this requires elaborate experimental measurements and is not routinely available. Equation (16.67) works satisfactorily in adiabatic boundary layers (Plate 1971). 

A polymer solution (density 1022 kg/m ) flows over the surface of a flat plate at a free stream velocity of 2.25 m/s. Estimate the laminar boundary layer thickness and surface shear stress at a point 300 mm downstream from tlie leading edge of the plate. Determine the total drag force on the plate from the leading edge to this point. What is the effect of doubling the free stream velocity ... 

The waU shear stress is an essential quantity of interest in a wall-bounded flow. The time averaged value of wall shear stress can be used to determine the skin friction drag acting on the body by the fluid flow. The time-resolved behavior of surface shear stress indicates the unsteady flow structures responsible for individual momentum transfer events and turbulence activities. The instantaneous shear stress values at distributed locations on the surface can be used to feedback control the turbulence events inside a boundary layer. Shear stress measurement also helps in assessment and control of power consumption rate and therefore is an important quantity of interest in various industrial applications. 

Equation 11.12 does not fit velocity profiles measured in a turbulent boundary layer and an alternative approach must be used. In the simplified treatment of the flow conditions within the turbulent boundary layer the existence of the buffer layer, shown in Figure 11.1, is neglected and it is assumed that the boundary layer consists of a laminar sub-layer, in which momentum transfer is by molecular motion alone, outside which there is a turbulent region in which transfer is effected entirely by eddy motion (Figure 11.7). The approach is based on the assumption that the shear stress at a plane surface can be calculated from the simple power law developed by Blasius, already referred to in Chapter 3. 

Thus, the shear stress is expressed as a function of the boundary layer thickness S and it is therefore implicitly assumed that a certain velocity profile exists in the fluid. As a first assumption, it may be assumed that a simple power relation exists between the velocity and the distance from the surface in the boundary layer, or ... 

If the velocity profile is the same for all stream velocities, the shear stress must be defined by specifying the velocity ux at any distance y from the surface. The boundary layer thickness, determined by the velocity profile, is then no longer an independent variable so that the index of < in equation 11.25 must be zero or ... 

If at a distance a from the leading edge the laminar sub-layer is of thickness 5 and the total thickness of the boundary layer is 8, the properties of the laminar sub-layer can be found by equating the shear stress at the surface as given by the Blasius equation (11.23) to that obtained from the velocity gradient near the surface. 

Assume that within the boundary layer outside the laminar sub-layer the velocity of flow is proportional to the one-seventh power of the distance from the surface and that the shear stress R at the surface is... 

When a fluid flows past a solid surface, the velocity of the fluid in contact with the wall is zero, as must be the case if the fluid is to be treated as a continuum. If the velocity at the solid boundary were not zero, the velocity gradient there would be infinite and by Newton s law of viscosity, equation 1.44, the shear stress would have to be infinite. If a turbulent stream of fluid flows past an isolated surface, such as an aircraft wing in a large wind tunnel, the velocity of the fluid is zero at the surface but rises with increasing distance from the surface and eventually approaches the velocity of the bulk of the stream. It is found that almost all the change in velocity occurs in a very thin layer of fluid adjacent to the solid surface ... 

As the fluid s velocity must be zero at the solid surface, the velocity fluctuations must be zero there. In the region very close to the solid boundary, ie the viscous sublayer, the velocity fluctuations are very small and the shear stress is almost entirely the viscous stress. Similarly, transport of heat and mass is due to molecular processes, the turbulent contribution being negligible. In contrast, in the outer part of the turbulent boundary layer turbulent fluctuations are dominant, as they are in the free stream outside the boundary layer. In the buffer or generation zone, turbulent and molecular processes are of comparable importance. 

To model this, Duncan-Hewitt and Thompson [50] developed a four-layer model for a transverse-shear mode acoustic wave sensor with one face immersed in a liquid, comprised of a solid substrate (quartz/electrode) layer, an ordered surface-adjacent layer, a thin transition layer, and the bulk liquid layer. The ordered surface-adjacent layer was assumed to be more structured than the bulk, with a greater density and viscosity. For the transition layer, based on an expansion of the analysis of Tolstoi [3] and then Blake [12], the authors developed a model based on the nucleation of vacancies in the layer caused by shear stress in the liquid. The aim of this work was to explore the concept of graded surface and liquid properties, as well as their effect on observable boundary conditions. They calculated the hrst-order rate of deformation, as the product of the rate constant of densities and the concentration of vacancies in the liquid. 

Here A a represents the difference between the interfacial tension at the end and at the beginning of the path. When the refreshing of the elements of liquid is complete, A a is equal to the difference between the interfacial tension at the equilibrium concentration at the interface and the interfacial tension between the liquid phases at their bulk concentrations. The problem of the boundary layer that develops when a solid planar surface moves continuously was treated by several authors. Tsou et al. [115] have derived the following expression for the local wall shear stress r ... 

The shearing stress, r, exerted by the wind on the ground entails a downwards flux of momentum. In the aerodynamic boundary layer above the surface, the momentum is transferred by the action of eddy diffusion on the velocity gradient. The friction velocity is defined by w = t/pa and is a measure of the intensity of the turbulent transfer. Near to a rough surface, the production of turbulance by mechanical forces... 

Vapor can condense on a cooled surface in two ways. Attention has mainly been given in this chapter to one of these modes of condensation, i.e.. to him condensation. The classical Nusselt-type analysis for film condensation with laminar film flow has been presented hnd the extension of this analysis to account for effects such as subcooling in the film and vapor shear stress at the outer edge of the film has been discussed. The conditions under which the flow in the film becomes turbulent have also been discussed. More advanced analysis of laminar film condensation based on the use of the boundary layer-type equations have been reviewed. 

Consider laminar film condensation on a vertical plate when the vapor is flow ing parallel to the surface in a downward direction at velocity, V. Assume that a turbulent boundary layer is formed in the vapor along the outer surface of the laminar liquid film. Determine a criterion that will indicate when the effect of the shear stress at the outer edge of the condensed liquid film on the heat transfer rate is less than 59c. Assume that pv [Pg.602]

Hydrodynamic boundary layer — is the region of fluid flow at or near a solid surface where the shear stresses are significantly different to those observed in bulk. The interaction between fluid and solid results in a retardation of the fluid flow which gives rise to a boundary layer of slower moving material. As the distance from the surface increases the fluid becomes less affected by these forces and the fluid velocity approaches the freestream velocity. The thickness of the boundary layer is commonly defined as the distance from the surface where the velocity is 99% of the freestream velocity. The hydrodynamic boundary layer is significant in electrochemical measurements whether the convection is forced or natural the effect of the size of the boundary layer has been studied using hydrodynamic measurements such as the rotating disk electrode [i] and - flow-cells [ii]. 

A boundary layer is a region of a fluid next to a solid that is dominated by the shearing stresses originating at the surface of the solid such layers arise for any solid in a fluid, such as a leaf in air. Adjacent to the leaf is a laminar sublayer of air (Fig. 7-6), where air movement is predominantly parallel to the leaf surface. Air movement is arrested at the leaf surface and has increasing speed at increasing distances from the surface. Diffusion... 

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